Teks Book Mathematics KSSM (F4)

DUAL LANGUAGE PROGRAMME
FORM
MATHEMATICS FORM


kurikulum standard sekolah menengah
MATHEMATICS FORM 4
authors
Yeow Pow Choo Thavamani A/P Renu Kamalah A/P Raman
Wong Jin Wen
Vincent De Selva A/L Santhanasamy
translators
Lien Poh Choo Chang Tze Hin Chew Lee Kian Kho Choong Quan
editors
Premah A/P Rasamanie Cynthia Cheok Ching Tuing Tiew Eyan Keng
Tan Swee Chang
designer
Ardi Bin Lidding
illustrators
Asparizal Bin Mohamed Sudin Mohammad Kamal Bin Ahmad
Sasbadi Sdn. Bhd. 198501006847 (139288-X) (Wholly-owned subsidiary of Sasbadi Holdings Berhad 201201038178 (1022660-T))
2019


KEMENTERIAN PENDIDIKAN MALAYSIA
Book series no: 0178
KPM2019 ISBN 978-983-77-1531-8 First Published 2019
© Ministry of Education Malaysia
All right reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, either electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Director General of Education Malaysia, Ministry of Education Malaysia. Negotiations are subject to an estimation of royalty or an honorarium.
Published for Ministry of Education Malaysia by:
Sasbadi Sdn. Bhd. 198501006847 (139288-X) (Wholly-owned subsidiary of Sasbadi Holdings Berhad 201201038178 (1022660-T))
Lot 12, Jalan Teknologi 3/4,
Taman Sains Selangor 1, Kota Damansara, 47810 Petaling Jaya,
Selangor Darul Ehsan, Malaysia.
Tel: +603-6145 1188 Fax: +603-6145 1199 Website: www.sasbadisb.com
E-mail: enquiry@sasbadi.com
Layout and Typesetting:
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Font size: 11 point
Printed by:
Dicetak di Malaysia oleh
C.T. Book Makers Sdn. Bhd. (416129-H) Lot 530 & 531, Jalan Perusahaan 3, Bandar Baru Sungai Buloh,
47000 Selangor.
ACKNOWLEDGEMENT
The publishing of this textbook involves cooperation from various parties. Our wholehearted appreciation and gratitude goes out to all involved parties:
• Educational Resources and Technology Division
• Committee members of Penyemakan Naskhah Sedia Kamera, Educational Resources and Technology Division, Ministry of Education Malaysia.
• Committee members of Penyemakan Pembetulan Naskhah Sedia Kamera, Educational Resources and Technology Division, Ministry of Education Malaysia.
• Officers of the Educational Resources and Technology Division and the Curriculum Development Division, Ministry of Education Malaysia.
• Officers of the English Language Teaching Centre (ELTC), Institute of Teacher Education Malaysia, Ministry of Education Malaysia.
• Chairperson and members of the Quality Control Panel.
• Editorial Team and Production Team, especially the illustrators and designers.
• Everyonewhohasbeendirectlyorindirectly involved in the successful publication of this book.


Contents
Introduction v Symbols and Formulae vii
1
1.1 Quadratic Functions and Equations 2 32
2.1 Number Bases 34 54
3.1 Statements 56 3.2 Arguments 71
94
4.1 Intersection of Sets 96 4.2 Union of Sets 106 4.3 Combined Operations on Sets 116
128
5.1 Network 130 Saiz seb
Quadratic Functions and Equations in One Variable
CHAPTER 1
CHAPTER 2
Number Bases
CHAPTER 3
Logical Reasoning
CHAPTER 4
Operations on Sets
CHAPTER 5
Network in Graph Theory
iii
enar


CHAPTER 6
Linear Inequalities in Two Variables
CHAPTER 7
Graphs of Motion
CHAPTER 8
Measures of Dispersion for Ungrouped Data
CHAPTER 9
CHAPTER 10
Answers Glossary References Index
se
154
6.1 Linear Inequalities in Two Variables 156
6.2 Systems of Linear Inequalities in Two Variables 165
182
7.1 Distance-Time Graphs 184
7.2 Speed-Time Graphs 195
210
8.1 Dispersion 212
8.2 Measures of Dispersion 219
242
9.1 Combined Events 244
9.2 Dependent Events and Independent Events 246
9.3 Mutually Exclusive Events and Non-Mutually Exclusive Events 253
9.4 Application of Probability of Combined Events 262
270
10.1 Financial Planning and Management 272
293
309
311
312
Probability of Combined Events
Consumer Mathematics: Financial Management
Download a free QR Code scanner application to your mobile device.
Note: Pupils can download a free dynamic geometry software to open the related files.
benar
http://bt.sasbadi.com/m4iv
Saiz
iv


Introduction
This Form 4 Mathematics Textbook is prepared based on Kurikulum Standard Sekolah Menengah (KSSM). This book contains 10 chapters arranged systematically based on Form 4 Mathematics Dokumen Standard Kurikulum dan Pentaksiran (DSKP).
At the beginning of each chapter, pupils are introduced to materials related to daily life to stimulate their thinking about the content. The learning standards and word lists are included to provide a visual summary of the chapter’s content.
special features of this book are:
Why Study This Chapter?
Walking Through Time WORD BANK
Mind Stimulation INFO ZONE
TIPS
You will learn
contains learning standards that pupils need to achieve in each chapter
tells the importance of knowledge and skills to be learned in this chapter
contains the historical background or origin of the content
contains key vocabulary in each chapter
contains activities that help pupils understand the basic mathematical concepts
contains additional information about the content
contains additional knowledge that pupils need to know
contains additional facts and common mistakes that pupils need to know
description
Indicator
Saiz seb
v
enar


Smart Mind
INTERACTIVE ZONE
MY MEMORY
Malaysiaku PROJECT
2.1a
description
contains challenging tasks to stimulate pupils’ critical and creative thinking skills
develops pupils’ mathematical communication skills
helps pupils to recall what they have learned
contains mathematical concepts related to Malaysia's achievements
enables pupils to carry out and present project work
assesses pupils’ understanding on the concepts they have learned
contains questions of various thinking skill levels
enables pupils to scan a QR Code using a mobile device for further information
covers the use of digital tools, calculators, hands-on activities and games to enhance pupils’ understanding more effectively
summarises the chapter
guides pupils to self-assess their achievement
contains alternative methods to check the answers
contains questions to test pupils’ higher order thinking skills
Mathematics Exploration
CONCEPT MAP
Self Reflection
Checking Answer
Saiz sebenar
vi


Symbols and Formulae
symbols
=
≠
≈
>
<
≥ ≤
~
p ⇒ q p⇔ q
∈ an element of
∉ not an element of
ξ universal set
⊂ a subset of
⊄ not a subset of mean
is equal to
is not equal to
is approximately equal to is more than
is less than
is more than or equal to is less than or equal to (tilde) negation
if p, then q
p if and only if q
n(A) number of elements of set A Σ sum
σ2 variance
σ standard deviation
A complement of set A { }, φ empty set
∩ intersection
∪ union
G graph e edge
v vertex d degree
Formulae
n(A < B) = n(A) + n(B) – n(A > B) n(A) = n(ξ) – n(A)
n(A > B ) = n(A < B) n(A < B ) = n(A > B)
P(A) = n(A) n(S )
Complement of event A, P(A' ) = 1 – P(A) P(A and B) = P(A > B)
P(A > B) = P(A) × P(B)
P(A or B) = P(A < B)
P(A < B) = P(A) + P(B) – P(A > B) Σd(v) = 2E; v ∈ V
y-intercept m = – —–———— x-intercept
Distance Speed = ——
Vertical distance Gradient, m = —————————–
Standard deviation,
Σf Σf Σ(x – )2 Σ x2
y–y 21
N
m = ———
Standard deviation,
22
Σx = —–
N
Σfx = —–
Time
Total distance travelled Average speed = ———————————
Change of speed Acceleration = ————–———
Σf
Variance, σ2 = ———— = —– –
Σf(x – )2 Σ fx2 Variance, σ2 = ————– = —– –
Total time taken
Change in time
Σ(x – )2 Σ x2 NN
2
2
Horizontal distance
σ =
———— = —– –
2
N
Σ f(x – ) Σfx
x–x 21
σ =
————– = —– – Σf Σf
2
Saiz s
eb
vii
enar


CHAPTER
1
Quadratic Functions and Equations
in One Variable
You will learn
► Quadratic Functions and Equations
Pulau Warisan is located in Kuala Terengganu. The island becomes a new tourist attraction because it is a man-made island connected with a bridge. This bridge is similar to the one in Putrajaya.
Do you know that the shape of this bridge has special mathematics characteristics?
Saiz sebenar
Why Study This Chapter?
Quadratic functions and equations are widely used in science,
business, sports and others. In sports, quadratic functions are
important in sports events such as shot put, discus and javelin.
In architecture, we often see curved structures in the shape of
parabola which are related to the mastery of quadratic concepts.
2


Chapter 1 Quadratic Functions and Equations in One Variable
WORD BANK
Walking Through Time
Al-Khawarizmi
(780 AD – 850 AD)
Al-Khawarizmi is well-known as the Father of Algebra. He was the founder of a few mathematics concepts. His work in algebra was outstanding. He was responsible for initiating the systematic and logical approach in solving linear and quadratic equations.
• quadratic function • axis of symmetry • variable
• root
• maximum point • minimum point
• fungsi kuadratik • paksi simetri
• pemboleh ubah • punca
• titik maksimum • titik minimum
http://bt.sasbadi.com/m4001
r
S
S
a
ai
iz
z
s
s
e
e
1
1
b
be
en
n
a
ar
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
1.1
Quadratic Functions and Equations
What is a quadratic expression in one variable?
Have you ever sketched the movement of a ball kicked by a football player, as shown in the picture?
The shape of this movement is a parabola.
Do you know that this parabola has its own equation, just like a straight line which has its own equation?
Learning Standard
Identify and describe the characteristics of quadratic expressions in one variable.
Mind Stimulation 1
Aim: To identify and describe the characteristics of quadratic expressions in one variable.
Steps:
1. Based on the table in Step 3, insert all the expressions one by one in the dynamic geometry software as shown below.
2. Observe the graph obtained.
3. Complete the table below.
(a) x2 + 4x + 1
(b) x–2 – 1
(c) –2x2 – 2x + 5
(d) 5x + 4
(e) 3x2 – 2
(f) –2x2 + 4x
(g) x3 + 1
Discussion:
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing
Characteristic
Expression
Coordinates of the lowest or highest point (if any)
Shape of graph
or and has the highest point or the lowest point. Which expression is a quadratic expression? Justify your answer.
The graph of a quadratic expression is either
benar
Saiz se
2
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable From the activity in Mind Stimulation 1, it is found that:
In general,
A quadratic expression in one variable is an expression whereby the highest power for the variable is two.
Where;
a, b and c are constants and a ≠ 0, x is a variable.
For example:
The general form of a quadratic expression is ax2 + bx + c. INTERACTIVE ZONE
TIPS
TIPS
1
x2 + 2x – 1, –y2 + 3y, 3 m2 – m + 4 and 2n2 + 5
are quadratic expressions.
Why is a ≠ 0 in a quadratic expression? Discuss.
Besides x, other letters can be used to represent variables.
1
Determine whether each of the following expressions is a quadratic expression in one variable. If not, justify your answer.
(a) 2x2 + 5
(c) 3x2 + 2y + 1
(e) 2x2 – x3 2
Solution:
(b) x3 – 6 (d) 12m2
(f) 4x2 – x12
The values of constants b and c can be zero.
(a) 2x2 + 5 is a quadratic expression in one variable.
(b) x3 – 6 is not a quadratic expression because the highest power of the variable x is 3.
(c) 3x2 + 2y + 1 is not a quadratic expression in one variable because there are two variables, x and y.
(d) 12 m2 is a quadratic expression in one variable.
(e) 2x2 – x3 is not a quadratic expression because there is a variable with a power which is not
2
a whole number.
(f) 4x2 – x12 is not a quadratic expression because there is a variable with a power which is not
a whole number.
Saiz sebenar
MY MEMORY
x3 = 3 x – 2 2
x 12 = √ x
3
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
Mind Stimulation 2
Aim: To state the values of a, b and c in a quadratic expression.
Steps:
1. Observe (a) in the table below.
2. Determine the values of a, b and c for the subsequent quadratic expressions.
Quadratic expression
Comparison
(a)
2x2 – 3x + 1
2x2 – 3x + 1 ax2 +bx+c
a= 2 b= –3 c= 1
(b)
2x2 – 4
x2+ x+ ax2 +bx+c
a=b=c=
(c)
12 x 2 + 5 x – 32
a=b=c=
(d)
–x2 + x
a=b=c=
(e)
–x2 – 3x – 9
a=b=c=
(f)
12x2
a=b=c=
Discussion:
How do you determine the values of a, b and c?
From the activity in Mind Stimulation 2, it is found that:
All quadratic expressions can be written in the form of ax2 + bx + c, where a ≠ 0.
In a quadratic expression,
Saiz sebenar
INTERACTIVE ZONE
4
a is the coefficient of x2, b is the coefficient of x, c is a constant.
Why are a and b known as the coefficients and c the constant?
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
1.1a
1. Determine whether each of the following expressions is a quadratic expression in one variable. If not, justify your answer.
(a) x2 – 5 (b) 2x2 + x–2
(d) – 12m2 (e) x3 – x
11
(g) x2 + 4x – 1 (h) p2 – 2 p + 3
(c) 3y2 – 3x + 1 (f) x12 + 2x – 1
(i) n(n – 2) 2. Determine the values of a, b and c for each of the following quadratic expressions.
(a) 2x2 – 5x + 1 (d) – 12 p2 + 4p
(g) h2 + 32 h – 4
(b) x2 – 2x
(e) 1 – x – 2x2
(h) 13 k2 – 2
(c) 2y2 + 1 (f) 4x2
(i) 2r (r – 3)
What is the relationship between a quadratic function and many-to-one relation?
Learning Standard
Recognise quadratic function as many-to-one relation, hence, describe the characteristics of quadratic functions.
What is the difference between a quadratic expression and
a quadratic function?
MY MEMORY
Types of relation
• One-to-one relation
• One-to-many relation
• Many-to-one relation
• Many-to-many relation
A quadratic expression is written in the form of ax2 + bx + c, whereas a quadratic function
is written in the form of f(x)=ax2 +bx+c.
INTERACTIVE ZONE
Saiz sebenar
Discuss and give examples of many-to-one relation.
5
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
Mind Stimulation 3
INFO ZONE
For a quadratic function, y = f (x).
Aim: To recognise quadratic functions as many-to-one relation. Materials: Ruler, pencil
Steps:
1. Based on the graphs of functions f (x) below, draw a line which is parallel to the x-axis on
graphs (b) and (c), as in graph (a).
2. Mark the points of intersection between the graph of function f (x) and the straight line.
3. State the number of points of intersection and the coordinates of the points of intersection.
4. Repeat Steps 1 to 3 by placing the ruler at different values of f (x). Ensure the straight lines drawn are parallel to the x-axis.
(a) f (x) = x2 – 3x (b) f (x) = –x2 + 4x + 1 (c) f (x) = x2 – 3x + 2
4 f(x)
f(x) 6 f(x)
4 24
–2 O 2 4 –2
Points of intersection Points of intersection = = ( ), ( )
Discussion:
1. What is the relationship between the x-coordinates and y-coordinates of both points of intersection for each function?
2. What is the type of relation of a quadratic function?
x
2
x2 O24x
–2 O 2 4 Number of points
of intersection ===
Number of points of intersection
Number of points of intersection
2
Points of intersection = ( ), ( )
(4, 4), (–1, 4)
MY MEMORY
For a point on a Cartesian
plane, the x-coordinate
is the object and the
y-coordinate is the image. xy
4 –1
4
From the activity in Mind Stimulation 3, it is found that:
In general,
Saiz sebenar
6
All quadratic functions have the same image for two different objects.
The type of relation of a quadratic function is a many-to-one relation.
Scan the QR Code to watch the vertical line test. http://bt.sasbadi.com/m4006
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable What is the shape of the graph of a quadratic function?
From the activity in Mind Stimulation 4, it is found that:
TIPS
What is the maximum or minimum point of a quadratic function?
Each sketch of the graph of a quadratic function has the highest or lowest value of y-coordinate based on the shape of the sketch.
For the sketch of the graph of a quadratic function with a < 0, y1 is the highest value of y-coordinate and x1 is the corresponding value for y1. The point (x1, y1) is known as the maximum point.
For the sketch of the graph of a quadratic function with a > 0, y2 is the lowest value of y-coordinate and x2 is the corresponding value for y2.
The point (x2, y2) is known as the minimum point.
Saiz sebenar
Mind Stimulation 4
Aim: To identify and describe the relationship between the value of a and the shape of the graph of a quadratic function.
Steps:
1. Drag the slider slowly from left to right. Observe the shape of the graph.
2. Sketch at least two graphs for positive values of a and two graphs for negative values of a.
Discussion:
What is the relationship between the value of a and the shape of a graph?
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing/t5az2zwm
For a graph of f(x) = ax2 + bx + c, a ≠ 0
(a) there are only two shapes of the graphs,
(b) thevalueofadeterminestheshapeofthegraph.
a< 0
a> 0
f (x) y1
(x1, y1) O x1
Diagram 1 f (x)
x
y
O
(x2, y2) x x2
2
Diagram 2
The curved shape of the graph of a quadratic function is called a parabola.
7
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
Mind Stimulation 5
Aim: To explore the maximum or minimum point of a quadratic function.
Steps:
1. Based on the table in Step 2, insert the quadratic functions in the dynamic geometry software.
2. Complete the table below as in (a).
Value of a
Shape of graph
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing
Maximum / Minimum point and coordinates
Quadratic function
Maximum
(a) f (x) = – 1 x2 – 4x + 2 2
(b) f (x) = x2 – 4x + 3
(c) f (x) = –2x2 – 4x + 1
a = – 1 2
point Coordinates =
point Coordinates =
point Coordinates =
(– 4, 10)
3. Repeat Steps 1 and 2 for various quadratic functions.
Discussion:
What is the relationship between the value of a and the maximum or minimum point?
INFO ZONE
The maximum or minimum point is also called a stationary point or a turning point.
From the activity in Mind Stimulation 5, it is found that:
What is the axis of symmetry of the graph of a quadratic
function?
The axis of symmetry of the graph of a quadratic function is a straight line that is parallel to the y-axis and divides the graph into two parts of the same size and shape.
The axis of symmetry will pass through the maximum or minimum point of the graph of the function as shown in the diagram below.
For a quadratic function f (x) = ax2 + bx + c, the maximum point is obtained when a < 0, the minimum point is obtained when a > 0.
MY MEMORY
An axis of symmetry is a straight line that divides a geometrical shape or an object into two parts of the same size and shape.
Axis of symmetry
Smart Mind
The equation of the axis of symmetry for a quadratic function is
b
x = – 2a .
Maximum point
Saiz sebenar
8
Minimum point
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
Mind Stimulation 6
Aim: To draw and recognise the axis of symmetry of the graph of a quadratic function.
Steps:
1. Using a ruler, draw the axis of symmetry for each graph of
quadratic function below.
2. Write the equation of the axis of symmetry as in (a).
12 Equation of axis of symmetry
–1 x= 1
– 2x
(b) f (x) = 2x2 + 4x – 3 f (x)
(c) f (x) = –2x2 + 4x + 2 f (x)
(a) f (x) = x2 f (x)
1
O
Discussion:
1. What is the relationship between the axis of symmetry of the graph of a quadratic function
and the y-axis?
2. What is the relationship between the axis of symmetry of the graph of a quadratic function and the maximum or minimum point?
x
–2
–3 –2 –1 O 1 x –14
3 –3 2 –4 1
–5O12x Equation of axis of symmetry Equation of axis of symmetry
MY MEMORY
The equation of a straight line which is parallel to the y-axis is x = h.
From the activity in Mind Stimulation 6, it is found that:
In general,
The axis of symmetry of the graph of a quadratic function is parallel to the y-axis and passes through the maximum or minimum point.
Each graph of quadratic function has one axis of symmetry which passes through the maximum or minimum point.
Axis of symmetry
x = h
Maximum point (m, n)
Minimum point (h, k)
Axis of symmetry
x = m Saiz
sebenar
9
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
1.1b
1. Determine whether the shapes of the following graphs of quadratic functions is or . (a) f (x) = x2 – 4x + 1 (b) g (x) = –x2 + 2x – 4
2. For each graph of quadratic function f (x) = ax2 + bx + c below, state the range of value of a and state whether the graph has a maximum or minimum point.
(a) f (x) (b) f (x)
OxOx
3. Determine the maximum or minimum point and state the equation of the axis of symmetry for each graph of quadratic function below.
(a)
f (x) (b) 5
f (x) 10
5
O
(c)
f (x) 4
(d)
f (x)
•(0, 3)
O
–10 –15
(–4, 0)
x
2468 –5
246 –5
x
Saiz sebenar
–2 O x –4
10
2
• O•x
• (4, 3)
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable What are the effects of changing the values of a, b and c
on graphs of quadratic functions, f (x) = ax2 + bx + c?
Learning Standard
Investigate and make generalisation about the effects of changing the values of a, b and c on graphs of quadratic functions,
f(x)=ax2 +bx+c.
Mind Stimulation 7
Aim: To identify the effects of changing the values of a on graphs of quadratic functions f (x) = ax2 + bx + c.
Steps:
1. Drag the slider from left to right.
2. Observe the shape of the graph as the value of a changes.
Discussion:
What are the effects of changing the values of a to the graphs of quadratic functions?
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing/nhxfjgy3
From the activity in Mind Stimulation 7, it is found that:
The value of a determines the shape of the graph. In general,
INTERACTIVE ZONE
Discuss the effects on the curve of the graphs of quadratic functions when a < 0.
For the graph of a quadratic function f (x) = ax2 + bx + c, the smaller the value of a, the wider the curved shape of the graph and vice versa.
g(x) = a2x2 f (x) = a1x2 g(x) = –a2x2
a1 < a2 a1 < a2
f (x) = – a1 x2
Saiz sebenar
11
CHAPTER 1


Saiz se
Chapter 1 Quadratic Functions and Equations in One Variable
Mind Stimulation 8
Aim: To identify the effects of changing the values of b on graphs of quadratic functions f (x) = ax2 + bx + c.
Steps:
1. Drag the slider from left to right.
2. Observe the position of the axis of symmetry as the value of b changes.
Discussion:
What are the effects of changing the values of b to the graphs of quadratic functions?
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing/vpzgvwba
From the activity in Mind Stimulation 8, it is found that:
In general,
The value of b determines the position of the axis of symmetry.
For the graph of a quadratic function f (x) = ax2 + bx + c
if a > 0;
b > 0, then the axis of symmetry lies on the left of the y-axis. b < 0, then the axis of symmetry lies on the right of the y-axis. b = 0, then the axis of symmetry is the y-axis.
f (x) f (x) OxOx
b>0
b>0
b<0
b=0
b=0
f (x) O
f (x) O
x
if a < 0;
benar
f (x)
b > 0, then the axis of symmetry lies on the right of the y-axis. b < 0, then the axis of symmetry lies on the left of the y-axis. b = 0, then the axis of symmetry is the y-axis.
f (x)
b<0
OxOx
x
12
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
Mind Stimulation 9
Aim: To identify the effects of changing the values of c on graphs of quadratic functions f (x) = ax2 + bx + c.
Steps:
1. Drag the slider from left to right.
2. Observe the position of the y-intercept as the value of c changes.
Discussion:
What are the effects of changing the values of c to the graphs of quadratic functions f(x) = ax2 + bx + c?
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing/rv7njx84
MY MEMORY
The y-intercept is a point of a graph that intersects the y-axis.
From the activity in Mind Stimulation 9, it is found that:
The value of c determines the position of the y-intercept.
In general,
For the graph of a quadratic function f (x) = ax2 + bx + c, the value of c determines the y-intercept of the graph.
f (x) f (x)
a< 0
cxc OO
x
a> 0
2
The quadratic function f (x) = x2 – 3x + c passes through a point A as given below. Calculate the value of c for each of the following cases.
(a) A(0, 4) (b) A(–1, 3)
Solution:
(a) The point A(0, 4) lies on the y-axis, thus c = 4.
TIPS
(b) f (x) = x2 – 3x + c
Substitute the values of x = –1 and f (x) = 3 into the quadratic function. 3 = (–1)2 – 3(–1) + c
c = –1
ebenar
c is the y-intercept of a quadratic function
f (x) = ax2 + bx + c.
Saiz s
13
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
The diagram shows two graphs of quadratic functions, y = f (x) and y = g (x), drawn on the same axes. State the range of the values of p. Explain your answer.
Solution:
0 < p < 3.
Since the curve of the graph g (x) is wider, thus p < 3. For a graph with the shape , p > 0.
Thus, 0 < p < 3.
g(x) = px2 – 4
y O
f (x) = 3x2 – 4
x
1.1c
1. The quadratic functions below pass through the points stated. Calculate the value of c for each case.
(a) f (x) = x2 + 7x + c, passes through point (0, 5).
(b) f (x) = 2x2 – 4x + c, passes through point (2, –3).
(c) f (x) = –2x2 + x + c, y-intercept = 4.
2. The diagram on the right shows two graphs of quadratic functions, y = f (x) and y = g(x), drawn on the same axes. State the range of the values of p. Explain your answer.
3. The diagram on the right shows the graph of a quadratic function f (x) = kx2 + 6x + h. Point A(3, 14) is the maximum point of the graph of quadratic function.
(a) Given k is an integer where –2 < k < 2, state the value of k.
(b) Using the value of k from (a), calculate the value of h.
(c) State the equation of the quadratic function formed when the graph is reflected in the x-axis. Give your answer in the form of f (x) = ax2 + bx + c.
Saiz sebenar
y g (x) = – 4x2 + 3
O
f (x) = –px2 + 3
14
3
f (x)
A(3,14)
h
O
x
x
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable How do you form a quadratic equation based on a situation?
A quadratic function is written in the form of f (x) = ax2 + bx + c while a quadratic equation is written in the general form, ax2 + bx + c = 0.
Mr Ganesan plans to make two different types of cards for Mathematics Club activities. The measurements of the cards are shown in the diagram below.
Learning Standard
Form quadratic functions based on situations,
and hence relate to the quadratic equations.
Try to guess my age. First I multiply my age with my own age. Next 21 times my age is subtracted from it. The result is 72.
4
x cm (2x + 1) cm x cm
x cm
(a) Form a quadratic expression for the total area of the two cards, A cm2, in terms of x. (b) The total area of the two cards is 114 cm2. Form a quadratic equation in terms of x.
Solution:
(a) A = x2 + x(2x + 1) (b) 3x2 + x = 114 = x2 + 2x2 + x 3x2 + x – 114 = 0
= 3x2 + x
1. The diagram on the right shows a piece of land with a length of (x + 20) m and a width of (x + 5) m.
(a) Write a function for the area, A m2, of the land.
1.1d
(b) If the area of the land is 250 m2, write a quadratic equation in terms of x. Give your answer in the form of ax2 + bx + c = 0.
(x + 20) m
2. Aiman is 4 years older than his younger brother. The product of Aiman and his younger brother’s
ages is equal to their father’s age. The father is 48 years old and Aiman’s younger brother is
p years old. Write a quadratic equation in terms of p.
Saiz sebenar
15
(x + 5) m
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
What do you understand about the roots of a quadratic
equation?
The roots of a quadratic equation ax2 + bx + c = 0 are the values of the variable, x, which satisfy the equation.
Do you know how the roots of a quadratic equation are determined?
INTERACTIVE ZONE
What is the meaning of “satisfy an equation”? Discuss.
Mind Stimulation 10
Aim: To determine the values of a variable that satisfy a quadratic equation. Steps:
1. Divide the class into two groups, A and B.
2. Group A will complete the table below without using the dynamic geometry software.
3. Group B will carry out this activity using the dynamic geometry software. Type each
quadratic expression into the software. For each graph, determine the value of the quadratic expression for each given value of x.
4. Complete and determine the values of x that satisfy the quadratic equation in the table below.
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing
Value of x
Value of x2 – 3x + 2
Value of x
Value of x2 – 5x + 4
Value of x
Value of x2 – 2x + 1
0
x2 – 3x + 2 = 0 x2 – 5x + 4 = 0 02 – 3(0) + 2 = 2 0
x2 – 2x + 1 = 0
–2
101 –1
2020
3231
4642
x are 1, 2 x are x are
x2 + x – 2 = 0 x2 – 4x + 5 = 2
–2 0 –3 –1 1 –2 0 2 –1 130
benar 2 4 1 x are x are x are
x2 + 2x – 2 = 1
Value of x
Value of x2 + x – 2
Value of x
Value of x2 – 4x + 5
Value of x
Value of x2 + 2x – 2
Saiz se
16
Learning Standard
Explain the meaning of roots of a quadratic equation.
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable How can you determine the values of the variable that satisfy a quadratic equation?
From the activity in Mind Stimulation 10, it is found that:
What is the relationship between the roots of a quadratic equation and the positions of the roots?
Discussion:
(a) There are one or two values of the variable that satisfy a quadratic equation.
(b) The values of the variable that satisfy a quadratic equation are known as the roots of the quadratic equation.
Mind Stimulation 11
Aim: To explore the positions of the roots of a quadratic equation on the graph of a quadratic function, f(x) = 0.
Steps:
1. Drag the slider to observe the changes of the x-coordinate and y-coordinate on the graph.
2. The roots of quadratic equation x2 – x – 6 = 0 can be determined when y = 0. Drag the slider from left to right. Observe the coordinates of A.
3. Determine the position of point A when y is 0.
4. Mark the point on the above diagram.
Discussion:
What do you notice about the positions of the roots of a quadratic equation on the graph of
the quadratic function?
Saiz s
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing/bykrknjx
ebenar
17
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable From the activity in Mind Stimulation 11, it is found that:
The roots of a quadratic equation ax2 + bx + c = 0 are the points of intersection of the graph of the quadratic function f(x) = ax2 + bx + c and the x-axis, which are also known as the x-intercepts.
a >0
f (x) f (x)
root
root
××x ××x root root
a< 0
For each graph of quadratic equation below, mark and state the roots of the given quadratic equation. (a) 2x2 + 5x – 12 = 0 (b) –x2 + 3x + 4 = 0
f (x) f (x) 56
x4 –5 2
–4 –3 –2 –1 O 1 2
–10 O x
Solution:
(a) 2x2 + 5x – 12 = 0
(b) –x2 + 3x + 4 = 0 f (x)
x = 1.5
–4 –3 –2 –1 O 1 2
x = –4
x
–15
–1 1234 –2
f (x) 56
4 –5 2
x = –1
–10 O x
Saiz sebenar
The roots are – 4 and 1.5.
18
5
–15
x = 4
–1 1234
–2
The roots are –1 and 4.
CHAPTER 1


(a) 2x2 – 7x + 3 Solution:
(a) 2x2 – 7x + 3 When x = 1,
= 0; x = 1, x = 3 = 0
= 2(1)2
= 2 – 7 + 3
= –2 not the same
(b) 3x2
Right: 0
– 7x + 5 = 3; x = 1, x = 13
MY MEMORY
The roots of a quadratic equation are the values of x that satisfy the equation.
Checking Answer
Saiz sebenar
When x = 3,
Left:
2x2 – 7x + 3 = 2(3)2 – 7(3) + 3
= 18 – 21 + 3 = 0
same
– 7x + 3 = 0. Right:
0
– 7x + 3 = 0.
Right: 3
– 7x + 5 = 3. Right:
3
– 7x + 5 = 3.
Chapter 1 Quadratic Functions and Equations in One Variable
6
Determine whether each of the following values is a root of the given quadratic equation.
Left:
2x2 – 7x + 3
– 7(1) + 3
Thus, x = 1 is not a root of the equation 2x2
1. Press 2 , Alpha,
,,,
2. Press CALC Display x?
0. 3.Press1 =
Display 2x2 – 7x + 3 –2.
4. Press CALC Display x?
1. 5.Press3 =
2–
7
X,x,,, Alpha X + 3
Display
0.
2x2 – 7x + 3
Thus, x = 3 is a root of the equation 2x2
(b) 3x2 – 7x + 5 = 3 When x = 1,
Left:
3x2 – 7x + 5 = 3(1)2 – 7(1) + 5
= 3–7+ 5
= 1 not the same Thus, x = 1 is not a root of the equation 3x2
1 When x = 3,
Left: 1 1
3x2 – 7x + 5 = 3(3)2 – 7(3) + 5
= 13 – 73 + 5 = 3
same
Thus, x = 13 is a root of the equation 3x2
19
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
1.1e
1. For each graph of quadratic function below, state the roots of the given quadratic equation.
(a) 3x2
– 5x – 2 = 0 (b) –x2 + x + 20 = 0 f (x)
6
4 20 2 10
OxOx
–0.5
0.5 1 1.5 2
–4 –2
2 4 6
–2 –4
–10 –20
2. Determine whether each of the following values is a root of the given quadratic equation. (a) x2 – 5x + 6 = 0; x = 3, x = 2
(b) 2x2 – x – 1 = 0; x = 1, x = 12 1
(c) 3x2 – 5x – 2 = 0; x = – 3, x = –2 (d) 3x2 + 4x + 2 = 6; x = 2, x = 23
3. Determine whether each of the following values is a root of the given quadratic equation.
(a) (x – 1)(x + 4) = 0; x = –4, x = 2, x = 1
(b) 2(x – 3)(x – 5) = 0; x = –3, x = 3, x = 5
(c) 3(2 + x)(x – 4) = 0; x = –2, x = 2, x = 4
4. For the graph of quadratic function on the right, determine whether the given value of x is a root of the quadratic equation f (x) = 0.
f (x)
(1, 16)
f (x)
Saiz sebenar
–3 O
20
(a) x = 1
(b) x = –3 15
(c) x = 15
(d) x = 5
5
x
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable How do you determine the roots of a quadratic equation
Learning Standard
Determine the roots of a quadratic equation by factorisation method.
by factorisation method?
Factorisation method is one of the methods used to determine the roots of a quadratic equation.
A quadratic equation needs to be written in the form of ax2 + bx + c = 0 before we carry out factorisation.
Determine the roots of the following quadratic equations by factorisation method.
(a) x2 – 5x + 6 = 0
(b) x2 + 72x = 2
(c) x = 5x – 24 2 x –4
(d) (y + 2)(y + 1) = 2(y + 11)
Solution:
MY MEMORY
2x2 +5x–3
= (2x – 1)(x + 3)
7
INFO ZONE
A quadratic equation can also be solved by using: • method of completing
the squares. • formula
–b ± √b2 – 4ac x = ——————
2a
(a) x2 – 5x + 6 = 0 (x – 3)(x – 2) = 0 x = 3 or x = 2
(b) x 2 + 72 x = 2x2 + 7x =
=
=
=
2
4 0 0
5x – 24 x –4
Checking Answer
Steps to solve x2 – 5x + 6 = 0.
1. Press mode 3 times until the following display is shown.
EQN MAT VCT 123
2. Press 1 to choose EQN , which is
equation.
3. Display shows
unknowns? 2 3 press
4. Display shows Degree? 2 3
press 2 , for power of 2
5. Display shows a? Enter the value 1, then press =
6. Display shows b? Enter the value –5, then press =
7. Display shows c? Enter the value 6, then press =
8. x1 = 3 is displayed, press =
9. x2 = 2 is displayed.
Saiz sebenar
2x2 + 7x – 4 (2x – 1)(x + 4)
x = 12 or x = –4 x
(c)
x2
2
2(5x – 24) x2 – 4x = 10x – 48
x = 8 or x = 6
(d) (y + 2)(y + 1) = 2(y + 11)
y2 + 3y + 2 = 2y + 22
y2 + y – 20 = 0 (y + 5)(y – 4) = 0
y = –5 or y = 4
x (x – 4)
=
– 14x + 48 = 0 (x – 8)(x – 6) = 0
21
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
How do you determine the roots of a quadratic equation by the graphical method?
Mind Stimulation 12
Aim: To determine the roots of a quadratic equation on the graph of a quadratic function using the dynamic geometry software.
Steps:
1. Insert the quadratic equations in the dynamic geometry software.
2. Determine the roots of the quadratic equations and complete the following table.
Activity Sheet:
TIPS
Scan the QR Code to carry out this activity. https://www.geogebra. org/graphing
Quadratic Equation
Roots
(a) (b) (c) (d)
x2 – 9x + 18 = 0 4x2 + 4x – 3 = 0 –x2 + 9x – 20 = 0
x = 3, x = 6
The root of a quadratic equationax2 +bx+c=0 is the value of x which satisfies the quadratic equation.
–4x2 – 11x + 3 = 0
How do you determine the roots of a quadratic equation using the graphical method?
Discussion:
Saiz se
22
From the activity in Mind Stimulation 12, it is found that:
The roots of a quadratic equation ax2 + bx + c = 0 can be obtained using the graphical method by reading the values of x which are the points of intersection of the graph of the quadratic function f (x) = ax2 + bx + c and the x-axis.
a> 0
f (x) f (x)
××x ××x
O root root O benar
root root
a< 0
CHAPTER 1


2. Write each of the following quadratic equation.
equations in the general form. Hence, solve the quadratic
(a) m(m + 2) = 3
(d) a + 5a = 6
(g) (h – 2)(h – 1) = 12
(b) (e) (h)
3p(11 – 2p) = 15 8k = 2 + k
(2x – 1)2 = 3x – 2
(c) 12 y2 = 12 – y
(f) 2h + 6h = 7
(i) (r + 1)(r + 9) = 16r
How do you sketch the graphs of quadratic functions?
Chapter 1 Quadratic Functions and Equations in One Variable
1.1f
1. Determine the roots of each of the following quadratic equations using the factorisation method.
(a) x2 – 3x – 10 = 0 (d) 2x2 + 8x – 24 = 0 (g) –3x2 – x + 14 = 0
(b) (e) (h)
x2 – 10x + 16 = 0 2x2 + 3x – 9 = 0 x2 – 5x = 0
(c) 3x2 – 5x + 2 = 0 (f) 4x2 – 3x – 10 = 0 (i) x2 – 4 = 0
Learning Standard
Sketch graphs of quadratic functions.
When sketching the graph of a quadratic function, the following characteristics should be shown on the graph.
1 The correct shape of the graph.
2 y-intercept.
3 x-intercept or one point that passes through the graph.
Case 1
The graph of a quadratic function intersects the x-axis.
Sketch the following graphs of quadratic functions.
(a) f(x) = x2 – 4x + 3 (b) f (x) = x2 – 6x + 9 (c) f(x) = –x2 + 2x + 15 (d) f(x) = –2x2 + 18
Saiz sebenar
MY MEMORY
f (x) = x2 – 4x + 3
a = 1, b = –4, c = 3
8
MY MEMORY
The constant c of a quadratic function is the y-intercept of the graph of the quadratic function.
23
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
Solution:
(a) f (x) = x2 – 4x + 3
Value of a = 1 > 0, shape
Value of c = 3, y-intercept = 3 When f (x) = 0, x2 – 4x + 3 = 0 (x – 3)(x – 1) = 0
f (x) 3
x = 1 or x = 3 O13
(b) f (x) = x2 – 6x + 9
Value of a = 1 > 0, shape Value of c = 9, y-intercept = 9 When f (x) = 0, x2 – 6x + 9 = 0
(x – 3)(x – 3) = 0 x =3
(c) f (x) = –x2 + 2x + 15
Value of a = –1 < 0, shape
Value of c = 15, y-intercept = 15 When f (x) = 0, –x2 + 2x + 15 = 0
x2 – 2x – 15 = 0 (x – 5)(x + 3) = 0 x = –3 or x = 5
(d) f (x) = –2x2 + 18
Value of a = –2 < 0, shape
Value of b = 0, axis of symmetry is the y-axis Value of c = 18, y-intercept = 18
When f (x) = 0, –2x2 + 18 = 0
f (x) 9
O
15
O
3
x
x f (x)
–3
x
5
f (x) 18
Saiz sebenar
24
x2 – 9 = 0 (x + 3)(x – 3) = 0
x = –3 or x = 3
–3 O 3
x
CHAPTER 1


9
Sketch each of the following graphs of quadratic functions. (a) f (x) = x2 + 1
(b) f (x) = –x2 – 3 Solution:
(a) f (x) = x2 + 1
Value of a = 1 > 0, shape
Value of b = 0, axis of symmetry is the y-axis Value of c = 1, y-intercept is 1
thus the minimum point is (0, 1)
When x = 2, f (2) = 22 + 1 =5
(b) f (x) = –x2 – 3
Value of a = –1 < 0, shape
Value of b = 0, axis of symmetry is the y-axis Value of c = –3, y-intercept is –3
thus the maximum point is (0, –3)
When x = 1, f (1) = – (1)2 – 3 = –4
f (x) 5
(2, 5) 2
(1, – 4)
Chapter 1 Quadratic Functions and Equations in One Variable
Case 2
The graph of a quadratic function does not intersect the x-axis.
1. Sketch each of the following graphs of quadratic functions.
(a) f (x) = 2x2 + 2x – 24
(b) f (x) = x2 – 8x + 16
(c) f (x) = –2x2 + 2x + 40
(d) f (x) = –2x2 + 8
2. Sketch each of the following graphs of quadratic functions.
(a) f (x) = x2 + 5 (b) f (x) = 2x2 + 1 (c) f (x) = –x2 + 2
Saiz sebenar
1
O f (x)
1
O
–3 – 4
x
x
MY MEMORY
(a) f(x) = x2 + 1
a = 1, b = 0, c = 1
(b) f(x) = –x2 – 3
a = –1, b = 0, c = –3
MY MEMORY
If b = 0 for a quadratic function, then the y-axis is the axis of symmetry of the graph of the quadratic function.
1.1g
25
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable How do you solve problems involving quadratic
equations?
Joseph wants to make the framework of a box in the shape of a cuboid using wooden rods. The price of the wooden rod is RM5 per metre. The base of the cuboid is a square. The height of the cuboid is 30 cm more than the length of its base. The total surface area of the box is 4 800 cm2. Joseph’s budget to build the frame of a box is RM15. Determine whether Joseph has enough budget.
Saiz se
Solution:
Understanding the problem
Length of the base = x cm
Height of the cuboid = (x + 30) cm Total surface area = 4 800 cm2 Budget = RM15 for a box
x cm
Planning a strategy
Total surface area = 2(x)(x) + 4(x)(x + 30) = 2x2 + 4x2 + 120x
= 6x2 + 120x
6x2 + 120x = 4 800 6x2 + 120x – 4 800 = 0
x2 + 20x – 800 = 0
(x + 40)(x – 20) = 0
x = –40 or x = 20
x = –40 is not acceptable, thus x = 20 cm
The measurements of the box are 20 cm × 20 cm × 50 cm. Total length of the edges of the box = 8 × 20 cm + 4 × 50 cm
INFO ZONE
Actual cost = 3.6 × RM5 = RM18
The actual cost for a box is RM18.
= 160 cm + 200 cm = 360 cm
= 3.6 m
Checking Answer
The measurement of the length cannot be negative.
When x = 20
Area = 6(20)2 + 120(20)
= 2 400 + 2 400 = 4 800
Conclusion
benar
Joseph does not have enough budget to build the framework of the box.
26
Learning Standard
Solve problems involving quadratic equations.
10
• Determine the expression for the surface area of the cuboid.
• Form a quadratic equation.
• Solve the quadratic equation.
• Determine the measurements of the box
and the actual cost.
(x + 30) cm
CHAPTER 1
Implementing the strategy


Chapter 1 Quadratic Functions and Equations in One Variable
1.1h
1. A rectangular field needs to be fenced up using mesh wire. The length of the field is (5x + 20) m and its width is x m.
(a) Express the area of the field, A m2, in terms of x.
(b) Given the area of the field is 5 100 m2, calculate the cost of fencing the field if the cost of the mesh
wire used is RM20 per metre.
xm
2. Encik Kamarul drove his car at an average speed of (20t – 20) km h–1 for (t – 3) hours along a highway. The distance travelled by Encik Kamarul was 225 km. The highway speed limit is 110 km h–1. Did Encik Kamarul follow the highway speed limit?
1. Determine whether each of the following expressions is a quadratic expression in one variable. (a) p2 – 4p + 1 (b) 12y2 – 4y + 9 (c) 13 – 2b + a2
(d) –m + 1 (e) b2 + 2 (f) a2 + 2a + 1 3
2. State the equation of the axis of symmetry for each graph of quadratic function below. (a) f(x) (b) f(x)
(5x + 20) m
–2O 6x O x 3. Solve each of the following quadratic equations.
(j) h – 1 = 1
3 h+ 1
(b) x2 – 81 = 0
(e) 2x2 – x – 10 = 0 (h) 2p2 – 13p + 20 = 0
(c) y2 – 4y = 0
(f) (x – 2)2 = 16
(i) (k – 4)(k – 1) = 18
(a) 4x2 – 1 = 0
(d) x2 + 3x + 2 = 0 (g) m2 + 3m – 4 = 0
5. Show that the quadratic equation (m – 6)2 =12 – 2m can be written as m2 – 10m + 24 = 0.
(k) 2(x – 2)2 = 5x – 7
4. Given one of the roots of the quadratic equation x2 + px – 18 = 0 is 2, calculate the value of p.
Hence, solve the equation (m – 6)2 = 12 – 2m.
Saiz sebenar
(–1, 4)
(7, 4)
27
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
6. Determine the coordinates of the minimum point from the graph of the quadratic function
f (x) = x2 – 6x + 5.
7. Given x = 4 is the axis of symmetry of the graph of the quadratic function f (x) = 7 + 8x – x2,
determine the coordinates of the maximum point.
8. The diagram shows part of the graph of the quadratic f (x) function f (x) = –x2 + 6x – 5. The straight line AB is parallel
to the x-axis. Determine
(a) the coordinates of point A, O
P
x
B f (x)
(d) n
10. The diagram shows part of the graph of the quadratic f (x) function f(x) = a(x – h)(x – k) where h < k. Point P is the
minimum point of the graph of the quadratic function.
(b) the equation of the axis of symmetry,
(c) the coordinates of point B,
(d) the coordinates of the maximum point P.
A
9. The diagram shows the graph of the quadratic function f(x) = ax2 + 8x + c. Calculate the value of each of the following.
6 (c) a –3(m,n)–1 O
(a) c (b) m
x
x
12. Diagram 1 shows an isosceles triangle with a base of 4y cm and a height of (y + 5) cm. Diagram 2 shows a square with sides of y cm.
The area of the triangle is more than the area of the square by 39 cm2. Calculate the difference in perimeter between both shapes.
15
(b) Determine the equation of the axis of symmetry. O1P5
(a) Calculate the value of
(i) h, (ii) k, (iii) a.
(c) State the coordinates of point P.
11. The length of a rectangle is (x + 1) cm and its width is 5 cm less than its length.
(a) Express the area of the rectangle, A cm2, in terms of x.
(b) The area of the rectangle is 24 cm2. Calculate the length and width of the rectangle.
Saiz sebenar
4y cm Diagram 1
Diagram 2
28
(y + 5) cm
y cm
CHAPTER 1


13. The diagram shows a rectangular garden ABCD. E and F are two points on CD and AD respectively such thatCE=DF=xm.Thelengthsof AF=12mand DE = 15 m.
(a) Form an expression for the area of the rectangle, A m2, in terms of x.
(b) The area of the rectangle is 460 m2. Calculate the value of x.
(c) Aiman wants to build a small footpath from point E
to point F with tiles which costs RM50 per metre. AimanhasabudgetofRM1000,determinewhether B Aiman has enough budget to build the footpath.
E
x m
C
Chapter 1 Quadratic Functions and Equations in One Variable
A 12 m F x m D
15 m
14. The History Club of SMK Seri Jaya has drawn two rectangular murals in conjunction with Malaysia’s Independence Day.
(a) Express the difference in area between the two murals, A m2, in terms of x.
(b) The difference in area between the two murals is 10 m2. Calculate the value of x. (c) Calculate the perimeter of the smaller mural.
PROJECT
(3x + 1) m
(2x – 1) m
(x – 3) m
(x – 1) m
Use your creativity to build different shapes based on the examples below. Display your work at the Mathematics Corner.
Materials:
1. Graph paper/blank paper.
2. Protractor, a pair of compasses.
3. Coloured pens.
Saiz sebenar
29
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable CONCEPT MAP
Quadratic Functions and Equations in One Variable
Quadratic Expression
(a) Highest power is 2 General form ax2 + bx + c
(b) Involves one variable
a, b and c
are constants, a ≠ 0
Shape of graph, a < 0 Axis of symmetry
b < 0 f (x)
O x
Axis of symmetry
Quadratic Function General form
f(x)=ax2 +bx+c
Axis of symmetry b < 0
Shape of graph, a > 0 Axis of symmetry
Axis of symmetry
b >0 OxOx
b =0
f (x) O
y-intercept cf ( x )
f (x)
f (x)
y-intercept f (x)
c
x
b > 0
f (x) O
x
Oxxx OO
Axis of symmetry b = 0 f ( x )
General form ax2 + bx + c = 0
Saiz sebenar
Quadratic Equation
The roots of a quadratic equation are the values of the variable that satisfy the equation
a > 0 f (x)
root O root x
The roots of a quadratic equation can be determined using
(a) factorisation method
(b) graphical method
a < 0 root
f (x)
root x
30
O
CHAPTER 1


Chapter 1 Quadratic Functions and Equations in One Variable
2.
Self Reflection
1.
4.
2.
3.
6.
5.
Across Down
2. The shape of the graph of a quadratic function.
3. The highest point of the graph of a quadratic function.
4. The lowest point of the graph of a quadratic function.
5. A function which its highest power is two.
1. The vertical axis that passes through the maximum or minimum point of the graph of a quadratic function.
2. A method used to determine the roots of a quadratic equation.
6. The values of variable that satisfy a quadratic equation.
The shape of the graph of a quadratic function is one of the most common shapes found in our daily life. Observe the following photos.
Use your creativity to draw a quadratic structure. Saiz seb
31
enar
CHAPTER 1


Chapter 2 Number Bases CH2APTER
Num
ber Bases
You will learn
► Number Bases
Malaysia has become a major focus of various technological developments which are capable of transforming the people’s lifestyle in this 21st century. These advances in technology enable Malaysians to enjoy fast download rates, hologram technology in education, medicine, industries, self-driving cars and more. A society that is proficient in information technology and telecommunication needs to be well versed in number bases as they have become the basis of all technologies.
Do you know the relationship between the number bases and technology?
S
Why Study This Chapter?
Number bases are the keys to all calculations in daily life. Among the fields involved are computer science and other areas that use information technology as the basis of research and development such as biotechnology, design technology, aerospace design,
Sa
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pharmacy and others.
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CHAPTER 2


Chapter 2 Number Bases
WORD BANK
Walking Through Time
Brahmagupta
(598 AD – 668 AD)
Brahmagupta was an astronomer from the state of Rajasthan in the north-west of India. He introduced the digit 0 to the number system which has become the basis for all the number bases used in olden times and today.
• number base • binary
• index
• place value
• digit value
• number system
• asas nombor • binari
• indeks
• nilai tempat • nilai digit
• sistem nombor
http://bt.sasbadi.com/m4033
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CHAPTER 2


Chapter 2 Number Bases
2.1
How do you represent and explain numbers in various bases in terms of numerals, place values, digit values and number values based on the collection process?
Number bases are number systems consisting of digits from 0 to 9. The number systems are made up of numbers with various bases. Base ten is a decimal number system used widely in daily life.
The table below shows the digits used in base two up to base ten.
Do you know which number base is used in computer science?
Number bases such as base 2, base 8, base 10 and base 16 are some of the number bases used in computer science.
Number base
Digit
INFO ZONE
Digits are the symbols used or combined to form a number in the number system. 0, 1, 2, 3, 4,
5, 6, 7, 8, 9 are the ten digits used in the decimal number system.
For example, 2 145 has
4 digits.
Saiz sebenar
Base 2 Base 3 Base 4 Base 5 Base 6 Base 7 Base 8 Base 9 Base 10
0, 1
0, 1, 2
0, 1, 2, 3
0, 1, 2, 3, 4
0, 1, 2, 3, 4, 5
0, 1, 2, 3, 4, 5, 6
0, 1, 2, 3, 4, 5, 6, 7
0, 1, 2, 3, 4, 5, 6, 7, 8 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
34
Number Bases
Learning Standard
Represent and explain numbers in various bases in terms of numerals, place values, digit values and number values based on the collection process.
CHAPTER 2


1
Give two examples of numbers that represent numbers in base two up to base ten.
Solution:
Chapter 2 Number Bases
Number base
Number
2
102
10012
3
213
12013
4
234
2134
5
415
3425
6
356
45106
7
647
4637
8
178
4728
9
789
3859
10
6910
289310
TIPS
Each base has digits from 0 to a digit which is less than its base. For example, base two has only digits 0 and 1.
INFO ZONE
number 325 base
is read as "Three two base five"
What are the place values involved in numbers in base two up to base ten?
Each base has place values according to each respective base. The place values of a base are the repeated multiplication of that base. Let’s say a is a base, then its place values start with a0, a1, a2, ..., an as shown in the table below.
MY MEMORY
a - base
n - power
a4 = a × a × a × a
an
Number base
an
Place value
a7
a6
a5
a4
a3
a2
a1
a0
Base 2
2n
128
64
32
16
8
4
2
1
Base 3
3n
2187
729
243
81
27
9
3
1
Base 4
4n
16384
4096
1024
256
64
16
4
1
Base 5
5n
78125
15625
3125
625
125
25
5
1
Base 6
6n
279936
46656
7776
1296
216
36
6
1
Base 7
7n
823543
117649
16807
2401
343
49
7
1
Base 8
8n
2097152
262144
32768
4096
512
64
8
1
Base 9
9n
4782969
531441
59049
6561
729
81
9
1
Base 10
10n
10000000
1000000
100000
10000
1000
100
10
S
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aebenar
35
CHAPTER 2


Chapter 2 Number Bases
2
State the place value of each digit in the numbers below. (a) 62318 (b) 1111012
Solution:
(a)
(b)
How do you state the value of a particular digit in a number in various bases?
The value of a particular digit in a number is the multiplication of a digit and the place value that represents the digit.
Walking Through Time
Konrad Zuse (1910 – 1995) was the inventor and pioneer of modern computers from Germany. He was the founder of the programmable computer. He designed the first high-level programming language known as Plankalkuel.
Number in base 8
6231
Place value
83 82 81 80
Number in base 2
111101
Place value
25 24 23 22 21 20
Multiplication of digit and place value
Use of place value block
1 0 1 0 23 22 21 20
2 012 33 32 31 30
4 432 53 52 51 50
10102
20123
44325
Saiz sebena
1 0 1 0 10102 23 22 21 20
2 012 20123 33 32 31 30
4 432 44325 53 52 51 50
Number
Number
Place value
Place value
Digit value
1 × 21 =2
Digit value
2
Number
Number
Place value
Place value
Digit value
2 × 33 = 54
Digit value
54
Number
Number
Place value
Place value
36
Digit value
r
4 × 53 = 500
Digit value
500
CHAPTER 2


Multiplication of digit and place value
2718 2 7 1 2718 82 81 80
State the value of the underlined digit in each of the following numbers.
2 7 1
Chapter 2 Number Bases Use of place value block
Number
Place value
Number
Place value
82
(d) 21345 (d) 21345
81 80
(a) 3418
Solution:
(a) 3418
3 × 82 = 192
(b) 50379 (b) 50379
5 × 93
= 3645
(c) 35016 (c) 35016
5 × 62
= 180
4 × 50
= 4
Digit value
2 × 82 = 128
Digit value
128
3
53
52
51
50
2
1
3
4
82
81
80
3
4
1
93
92
91
90
5
0
3
7
63
62
61
60
3
5
0
1
How do you state the numerical value of a number in various bases?
The numerical value of a number in various bases can be determined by calculating the sum of the digit values of the number.
(a) Determine the value of a number in base two.
Collection process
Indicator
Numbers in base two have only digits 0 and 1.
Number
Place value
Digit value
1 × 24 = 16
1 × 23 =8
0 × 22 =0
0 × 21 =0
1 × 20 =1
Number value
16 + 8 + 0 + 0 + 1 = 2510
11001
24 23 22 21 20
Saiz sebenar
37
CHAPTER 2


Chapter 2 Number Bases
Adding digit values using blocks
11001
24 23 22 21 20
16 + 8 + 0 + 0 + 1 = 2510
(b) Determine the value of a number in base three.
16
8
1
Collection process
12021
34 33 32 31 30 81 + 54 + 0 + 6 + 1 = 14210
1 × 34 = 81
2 × 33 = 54
0 × 32 =0
2 × 31 =6
1 × 30 =1
Adding digit values using blocks
12021
34 33 32 31 30
81
54
6
1
81 + 54 + 0 + 6 + 1 = 14210
(c) Determine the value of a number in base four.
INTERACTIVE ZONE
Collection process
Saiz se
3021
43 42 41 40 192 + 0 + 8 + 1 = 20110
Is the value of 2438 equal to 2435? Discuss.
INFO ZONE
Writing the base sign for a number in base 10 is optional, which can be written or left out.
3 × 43 = 192
0 × 42 =0
2 × 41 =8
1 × 40 =1
38
Number
Place value
Digit value
Number value
Number
Place value
Digit value
Number value
Number
Place value
Digit value
Number value
Number
Place value
Digit value
benar
Number value
CHAPTER 2


Adding digit values using blocks
3021
43 42 41 40
Smart Mind
State two numbers in different bases with the same value.
Smart Mind
Convert your year of birth to a number base that you prefer.
INTERACTIVE ZONE
Chapter 2 Number Bases
Number
Place value
Digit value
192
8
1
Number value
192+0+8+1=20110
Determine the values of the following numbers.
(a) 3405
Solution:
(a) 3405
(b) 3417
(c) 15068
(b) 3417
(c) 15068
4
Number
Place value
Number value
(3 × 52) + (4 × 51) + (0 × 50) = 75 + 20 + 0
= 9510
340
52 51 50
341
72 71 70
1506
83 82 81 80
Checking Answer
What will happen if
a number in a base higher than 10 is used? Discuss.
Number
Place value
Number value
(3 × 72) + (4 × 71) + (1 × 70) = 147 + 28 + 1
= 17610
1. Press the MODE key 2 times until
appears on the screen.
2. Press 3 to choose
BASE .
3. Press OCT.
4. Press 1506 then press = .
5. Press DEC , the answer
838 is displayed.
Saiz s
SD REG BASE 123
Number
Place value
Number value
(1 × 83) + (5 × 82) + (0 × 81) + (6 × 80) = 512 + 320 + 0 + 6
= 83810
ebenar
39
CHAPTER 2


Chapter 2 Number Bases
2.1a
1. Write three numbers to represent numbers in base two up to base nine.
2. Circle three numbers which do not represent numbers in base six.
245 332 461 212 371 829 345 123
3.
4. Determine the place value of the underlined digit in each of the following numbers. (a) 11100102 (b) 2145 (c) 60017 (d) 511406 (e) 12003 (f) 6839 (g) 23314 (h) 73218 (i) 52416 (j) 32215
5. Determine the value of the underlined digit in each of the following numbers.
234 673 336 281
Based on the four numbers above, identify and list all the numbers with the following bases. (a) Base five (b) Base seven (c) Base eight (d) Base nine
(a) 11102 (b) 3245 (c) 8739 (d) 2356 (f) 166237 (g) 11012 (h) 17768 (i) 2314
6. Determine the values of the following numbers in base ten.
(e) 21003 (j) 1111012
(e) 3647 (j) 11213
(a) 236 (b) 4258 (c) 1101012 (f) 334 (g) 1235 (h) 12178
7. Determine the values of p and q.
(a) 11012 = (1 × 2p) + (1 × q) + (1 × 20)
(b) 3758 = (3 × 8p) + (q × 81) + (5 × 80)
(c) 13214 = (1 × pq) + (3 × 42) + (2 × 41) + (1 × 40)
(d) 3389 (i) 5156
8. Calculate the sum of the values of digit 8 and digit 3 in 18239.
9. Rearrange the following numbers in ascending order.
(a) 1102, 11012, 1112, 11102 (b) 11234, 1324, 2314, 1124
10. Rearrange the following numbers in descending order. (a) 1111012, 12134, 819 (b) 1234, 738, 3135
(c) 3245, 1245, 2415, 2315 (c) 2536, 1617, 2223
11. Calculate the difference between the values of digit 5 in 15768 and 1257.
Saiz sebenar
40
CHAPTER 2


Chapter 2 Number Bases How do you convert numbers from one base to another base using
various methods?
A number can be converted to other bases by using various methods, such as the division using place value and the division using base value. These processes involve converting
(a) a number in base ten to another base.
(b) a number in a certain base to base ten and then to another base. (c) a number in base two directly to base eight.
(d) a number in base eight directly to base two.
How do you convert a number in base ten to another base?
A number in base ten can be converted to another base by dividing the number using the place value or the base value required. The number 5810 can be converted to base two by
(a) dividing 58 using the place value in base two.
(b) dividing 58 by two.
Rajang River which is the longest river in Malaysia is 563 kilometres. Convert 56310 to a number in
(a) base five. (b) base eight.
Solution:
(a) Base five
Division using place value
625 125 25 5 1
04223
Learning Standard
Convert numbers from one base to another using various methods.
5
Place value
The value of 625 is greater than 563
)4 125 563
– 500 63
)2 25 63
– 50 13
)2 5 13
– 10 3
)3 13
–3 0
Step
Base 5
Answer
42235
Alternative Method
Division using base value
563 is divided by the place value of 125. Its remainder is transferred to the previous place value for the next division until
a zero remainder is obtained.
5 563 5 112 5 22 5 4
– 3 – 2 – 2
Remainder
The digits are read from the bottom upwards.
0 – 4
Saiz sebenar
563 = 4223 The division is continued until digit zero is obtained. 10 5
41
CHAPTER 2